Theoretical Study of Electromagnetic Wave Propagation: Gaussian Bean Method ()
1. Introduction
Along with the rapid evolution of fiber optics, integrated optics and application of laser in both medicine and technology, there has been a growing interest in the study of Gaussian beam propagation. This is based on the fact that Gaussian beam has to do with focusing and modification of shape of propagating electromagnetic wave or laser beam. From the earlier finding on the research on laser beam, it has been found that laser beam propagation can be approximated by an ideal Gaussian beam intensity profile [1,2]. Understanding of the basic properties of Gaussian beam has been specifically found to be very vital. I select the best optics for practical application [3]. Sequel to this, lots of scientists had worked on Gaussian beam applications in electromagnetic wave propagation and in optics. For instance, a work has been carried out on nonspecular phenomena for beam reflection at monolayer and multilayered dielectric interface respectively from where it has revealed that under various conditions nonspecular beam phenomena is more realizable [4-6]. Tamir on his own presented a unified and simplified analysis of the lateral and longitudinal displacement with angular deflection on reflection of a Gaussian beam at a dielectric interface with two or more layer [7,8].
However in this paper, we intend to study electromagnetic wave propagation using the concept of Gaussian beam starting from general wave equation with which obtain the Gaussian function in which waves operating on the fundamental transverse mode is approximated to Gaussian profile. The distribution profile is analyzed with intent to observe the influence of the beam waist on the profile.
2. Theoretical Framework
In this case we start with wave equation in terms of electric field given as
(1)
where 
is the dielectric constant as a function wave solution of the form
(2)
Which if we consider medium with non-uniform refractive index, we have
(3)
where the propagation constant in the medium is given by
(4)
With this, Equation (2) becomes
(5)
In general, if the medium is absorbing or exhibit gain, the its dielectric constant 
is considered to have real and imaginary part, but in a situation where the medium conducts with conductivity 
then the complex propagation vector is introduce which obeys.
[9]. (6)
However, a simple solution of this type is inadequate to describe the field distributions of transverse mode. As a result, we seek solution to the plane wave equation of the form.
(7)
Propagating in the z-direction and localized the z-axis. Thus the idea is to obtain a solution of the wave equation that gives phase front that can be approximated over a narrow region. Thus substituting Equation (7) into Equation (4) we obtain.
(8)
Since we are looking for a paraxial beam-like solution, then 
varies slowly with 
and Equation (8) becomes
(9)
With a solution given as
(10)
where 
is the square distant of the points x, y from the axis of propagation where 
and 
are beam parameters Equation (10) gives the fundamental Gaussian beam of time-independent wave equation.
3. Intensity of Gaussian Beam
The intensity of 
of Gaussian beam is given by
(11)
where 
and 
are the complex conjugate of 
and 
respectively hence
(12)
Considering two real beam parameters that 
and 
relating to 
by depend on 
(13)
is that describes the axial distance from the Gaussian beam waist.
We can express Equation (11) as
(14)
(15)
Signifying the dependency of Gaussian beam intensity to r while w signifies the distance from the axis at 
where the intensity of the beam falls to 
of it its peak value on the axis 
total power of the beam with these parameters Equation (10) is now written as
(16)
is evaluate to give
(17)
where 
is a constant of integration the expresses the value of the beam parameter at the plane 
for which when we consider 
we obtain
(18)
where 
defines a constant of integration substituting Equation (18) into (16) we have
(19)
The factor 
is a constant phase factor. Thus
(20)
4. Diffraction Effect
The limitations of Gaussian beam based on the fact that even if the wave fronts were made flat at some plane, it quickly acquires curvature and begin to spread in accordance with
(21)
and
(22)
where 
is the distance propagated from the plane where the wave-front is flat 
wavelength of light,
is the radius of the 
irradiance contour at the plane where wave-front is flat, 
is the radius of the
contour after the wave has propagated a distance
.
Considering the optimum starting beam radius for a distance
, we have 
Equation (22) can be reduced to
(23)
With the irradiance distance distribution of the Gaussian beam described as
(24)
where 
and 
is the total power in the beam which is the same at all cross sections of the beams we also obtain that
(25)
(26)
at optimism starting beam radius for a given distance,
.
(27)
5. Discussion
From the displayed results in Figure 1, the distribution profile when y = 2, x = 2 and w = 2, the distribution beam width ranged within −5 to 5, In Figure 2, for y = 2 and x = 3 when w = 2, the width of the beam ranged from −6.5 to 6.5. In Figures 3 and 4 the distribution profiles displayed is found to be departed from normal distribution shape as in the other figures when w is increased to 5 for
Figure 1. Gaussian distribution profile when y = 2, w = 3.
Figure 2. Gaussian distribution profile when y = 2, w = 4.
Figure 3. Gaussian distribution profile when y = 2, w = 5.
y = 2 when w = 10, the departure became more pronounced pattern as shown in Figure 5. However, the distribution profile of percentage irradiance as function
Figure 4. Gaussian distribution profile when y = 2, w = 5.
Figure 5. Gaussian distribution profile when λ = 0.50 mm, y = 2, w = 10.
contour radius as displayed in Figure 6 indicate the fact that as the contour decreases, the beam distribution tappers showing that there is a point known as spot size where the intensity of the distribution is fallen to
.
This indicates that Gaussian distribution function depends on r as shown in Equation (16) that depicts characteristic of normal distribution function.
However, the shape of the distribution depends generally on diameter at which the intensity has fallen of 
as in Figure 7 when x = y coupled with the optimum beam waist that is given as
axial distance and Raleigh range which is given in Equation (22) with 
[9] These values provide the best combination for minimum starting beam diameter for good minimum spread according to [10] whose ratio is 
over a distant z and a particular wavelength.
Figure 8